Yakov Eliashberg

Introduction to Symplectic Field Theory

We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds in the spirit of topological field theory, and at the same time serves as a rich source of new invariants of contact manifolds and their Legendrian submanifolds. Moreover, we hope that the applications of SFT go far beyond this framework.1

TOPOLOGICAL CHARACTERIZATION OF STEIN MANIFOLDS OF DIMENSION >2

In this paper I give a completed topological characterization of Stein manifolds of complex dimension >2. Another paper (see [E14]) is devoted to new topogical obstructions for the existence of a Stein complex structure on real manifolds of dimension 4. Main results of the paper have been announced in [E13].

Compactness results in Symplectic Field Theory

This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in (4). We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromovs compactness theorem in (8) as well as compactness theorems in Floer homology theory, (6, 7), and in contact geometry, (9, 19).

Partially ordered groups and geometry of contact transformations

Abstract. We prove that, for a class of contact manifolds, the universal cover of the group of contact diffeomorphisms carries a natural partial order. It leads to a new viewpoint on geometry and dynamics of contactomorphisms. It gives rise to invariants of contactomorphisms which generalize the classical notion of the rotation number. Our approach is based on tools of Symplectic Topology.

Symplectic geometry of plurisubharmonic functions

In these lectures we describe symplectic geometry related to the notion of pseudo-convexity (or J-convexity). The notion of J-convexity, which is a complex analog of convexity, is one of the basic mathematical notions. Symplectic geometry built-in into this notion is essential for understanding the structure of affine (or Stein) complex manifolds. It plays also a major role in the classification of Stein complex structures up to deformation. Plurisubharmonic (or J-convex) functions on complex manifolds are analagous to convex functions on Riemannian manifolds but their theory is much richer. Symplectic geometry is crucial for understanding Morse-theoretic properties of J-convex functions.

LAGRANGIAN INTERSECTIONS IN CONTACT GEOMETRY

It is well-known that all problems of Contact geometry can be reformulated as problems of Symplectic geometry. This can be done via symplectization (see 2.1 below). In particular, the problem of Lagrangian intersections naturally arises in connection with several contact geometric questions (see 2.5 example, and below). However, there is one major difficulty when one tries to realize this approach: the symplectizations of contact manifolds are noncompact and, what is even worse, non-convex (see [EGr1]). This leads to the loss of compactness for the spaces of holomorphic curves and thus creates serious difficulties for the traditional Floer homology approach. The goal of this paper is to show that this problem can be successfully overcome by using an idea from [H].

The diameter of the symplectomorphism group is infinite

Shnirelman has proved that the group of volume-preversing diffeomorphisms of the cube in R 3 has finite diameter and has announced the result that this is false for the square. Despite the fact that Shnirelman formulated his theorem only for the 3-dimensional cube, his proof can be modified for the case of the group of volume preserving diffeomorphisms of any compact simply-connected Riemannian manifold of dimension > 2. It turns out that the situation with the group of symplectic diffeomorphisms is completely different. We prove that the diameter of the symplectomorphism group of any compact exact symplectic manifold (necessarily with boundary) is infinite

Symplectic homology product via Legendrian surgery.

This research announcement continues the study of the symplectic homology of Weinstein manifolds undertaken by the authors [Bourgeois F, Ekholm T, Eliashberg Y (2009) arXiv:0911.0026] where the symplectic homology, as a vector space, was expressed in terms of the Legendrian homology algebra of the attaching spheres of critical handles. Here, we express the product and Batalin–Vilkovisky operator of symplectic homology in that context.

Wrinkling of smooth mappings-II Wrinkling of embeddings and K. Igusa’s theorem

The method of wrinkling of singularities, described in our earlier paper [5], is applied in the current paper to prove a generalization of K. Igusa’s theorem about functions without higher singularities, as well as some related results. ( 2000 Elsevier Science Ltd. All rights reserved. MSC: 57R45

WRINKLING OF SMOOTH MAPPINGS III FOLIATIONS OF CODIMENSION GREATER THAN ONE

This is the third paper in our Wrinkling saga (see [EM1], [EM2]). The first paper [EM1] was devoted to the foundations of the method. The second paper [EM2], as well as the current one are devoted to the applications of the wrinkling process. In [EM2] we proved, among other results, a generalized Igusa’s theorem about functions with moderate singularities. The current paper is devoted to applications of the wrinkling method in the foliation theory. The results of this paper essentially overlap with our paper [ME], which was written twenty years ago, soon after Thurston’s remarkable discovery (see [Th1]) of an h-principle for foliations of codimension greater than one on closed manifolds. The paper [ME] contained an alternative proof of Thurston’s theorem from [Th1], and was based on the technique of surgery of singularities which was developed in [E2]. The proof presented in this paper is based on the wrinkling method. Although essentially similar to our proof in [ME], the current proof is, in our opinion, more transparent and easier to understand. Besides Thurston’s theorem we prove here a generalized version of our results from [ME] related to families of foliations.